= Reference-configuration jump balances
Let an interface move at signed speed $c$ in a reference unit normal direction $n$, and set $[a]=a^+-a^-$ with $n$ pointing from minus to plus. For continuous reference density and no surface sources, distributional <momentum conservation>, <conservation of energy> and the <Lagrangian entropy inequality> give
$$
\rho_0 c[v_i]+n_I[P_{Ii}]=0,\qquad
\rho_0 c[u+|v|^2/2]+n_I[P_{Ii}v_i-q_I^0]=0,
$$
$$
-\rho_0 c[\eta]+n_I[q_I^0/\theta]\geq0.
$$
For a level-set interface $\phi=0$, $\phi_t=-c|\nabla\phi|$, the singular part of $\partial_t a$ is $-c[a]\delta_S$ and that of $\operatorname{Div}b$ is $n\cdot[b]\delta_S$, proving these formulas. Using $V=-c$ reverses the speed terms. Density jumps require jumps of the complete conserved densities.
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